If the length of a simple pendulum is doubled then the `%` change in the time period is `:`

Options

A. 50
B. `41:4`
C. 25
D. 100

Explanation

To solve the problem, we need to determine the percentage change in the time period of a simple pendulum when its length is doubled.

### Step-by-Step Solution:

1. **Understand the formula for the time period of a simple pendulum**:
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Calculate the initial time period**:
Let the initial length of the pendulum be \( L \). Therefore, the initial time period \( T_1 \) is:
\[ T_1 = 2\pi \sqrt{\frac{L}{g}} \]

3. **Determine the new length when it is doubled**:
If the length is doubled, the new length \( L' \) is:
\[ L' = 2L \]

4. **Calculate the new time period**:
The new time period \( T_2 \) with the doubled length is:
\[ T_2 = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{2L}{g}} = 2\pi \sqrt{2} \sqrt{\frac{L}{g}} = \sqrt{2} \cdot (2\pi \sqrt{\frac{L}{g}}) \]
Thus, we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2 = \sqrt{2} \cdot T_1 \]

5. **Calculate the percentage change in time period**:
The percentage change in the time period can be calculated using the formula:
\[ \text{Percentage Change} = \frac{T_2 - T_1}{T_1} \times 100 \]
Substituting \( T_2 = \sqrt{2} \cdot T_1 \):
\[ \text{Percentage Change} = \frac{\sqrt{2} \cdot T_1 - T_1}{T_1} \times 100 \]
Simplifying this gives:
\[ \text{Percentage Change} = \frac{(\sqrt{2} - 1) \cdot T_1}{T_1} \times 100 = (\sqrt{2} - 1) \times 100 \]

6. **Calculate the numerical value**:
We know that \( \sqrt{2} \approx 1.414 \):
\[ \text{Percentage Change} = (1.414 - 1) \times 100 = 0.414 \times 100 = 41.4\% \]

### Final Answer:
The percentage change in the time period when the length of the pendulum is doubled is **41.4%**.